But, from another side, if the law were rigorously and always true, space would be very different from what it is. We should have categories strongly contrasted between which would be portioned out on the one hand the announcers A, on the other hand the parries B; these categories would be excessively numerous, but they would be entirely separated one from another. Space would be composed of points very numerous, but discrete; it would be discontinuous. There would be no reason for ranging these points in one order rather than another, nor consequently for attributing to space three dimensions.
But it is not so; permit me to resume for a moment the language of those who already know geometry; this is quite proper since this is the language best understood by those I wish to make understand me.
When I desire to parry the stroke, I seek to attain the point whence comes this blow, but it suffices that I approach quite near. Then the parry B1 may answer for A1 and for A2, if the point which corresponds to B1 is sufficiently near both to that corresponding to A1 and to that corresponding to A2. But it may happen that the point corresponding to another parry B2 may be sufficiently near to the point corresponding to A1 and not sufficiently near the point corresponding to A2; so that the parry B2 may answer for A1 without answering for A2. For one who does not yet know geometry, this translates itself simply by a derogation of the law stated above. And then things will happen thus:
Two parries B1 and B2 will be associated with the same warning A1 and with a large number of warnings which we shall range in the same category as A1 and which we shall make correspond to the same point of space. But we may find warnings A2 which will be associated with B2 without being associated with B1, and which in compensation will be associated with B3, which B3 was not associated with A1, and so forth, so that we may write the series
B1, A1, B2, A2, B3, A3, B4, A4,
where each term is associated with the following and the preceding, but not with the terms several places away.
Needless to add that each of the terms of these series is not isolated, but forms part of a very numerous category of other warnings or of other parries which have the same connections as it, and which may be regarded as belonging to the same point of space.
The fundamental law, though admitting of exceptions, remains therefore almost always true. Only, in consequence of these exceptions, these categories, in place of being entirely separated, encroach partially one upon another and mutually penetrate in a certain measure, so that space becomes continuous.
On the other hand, the order in which these categories are to be ranged is no longer arbitrary, and if we refer to the preceding series, we see it is necessary to put B2 between A1 and A2 and consequently between B1 and B3 and that we could not for instance put it between B3 and B4.
There is therefore an order in which are naturally arranged our categories which correspond to the points of space, and experience teaches us that this order presents itself under the form of a table of triple entry, and this is why space has three dimensions.